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Theorem trlnle 29643
Description: The atom not under the fiducial co-atom  W is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
trlne.l  |-  .<_  =  ( le `  K )
trlne.a  |-  A  =  ( Atoms `  K )
trlne.h  |-  H  =  ( LHyp `  K
)
trlne.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlne.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlnle  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )

Proof of Theorem trlnle
StepHypRef Expression
1 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  HL )
2 hlatl 28818 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 17 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  AtLat )
4 simpl3l 1012 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  P  e.  A )
5 trlne.l . . . . 5  |-  .<_  =  ( le `  K )
6 eqid 2285 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 trlne.a . . . . 5  |-  A  =  ( Atoms `  K )
85, 6, 7atnle0 28767 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  ( 0. `  K ) )
93, 4, 8syl2anc 644 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( 0. `  K ) )
10 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
12 simpl2 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  F  e.  T )
13 simpr 449 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  P )  =  P )
14 trlne.h . . . . . 6  |-  H  =  ( LHyp `  K
)
15 trlne.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
16 trlne.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
175, 6, 7, 14, 15, 16trl0 29627 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
1810, 11, 12, 13, 17syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( R `  F )  =  ( 0. `  K ) )
1918breq2d 4037 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  .<_  ( R `  F )  <->  P  .<_  ( 0. `  K ) ) )
209, 19mtbird 294 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( R `  F ) )
215, 7, 14, 15, 16trlne 29642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  ( R `  F ) )
2221adantr 453 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  =/=  ( R `  F
) )
23 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  HL )
2423, 2syl 17 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  AtLat )
25 simpl3l 1012 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  e.  A )
26 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( K  e.  HL  /\  W  e.  H ) )
27 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
28 simpl2 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  F  e.  T )
29 simpr 449 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( F `  P )  =/=  P )
305, 7, 14, 15, 16trlat 29626 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
3126, 27, 28, 29, 30syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( R `  F )  e.  A )
325, 7atncmp 28770 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( R `  F )  e.  A )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3324, 25, 31, 32syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3422, 33mpbird 225 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  -.  P  .<_  ( R `  F ) )
3520, 34pm2.61dane 2526 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   class class class wbr 4025   ` cfv 5222   lecple 13210   0.cp0 14138   Atomscatm 28721   AtLatcal 28722   HLchlt 28808   LHypclh 29441   LTrncltrn 29558   trLctrl 29615
This theorem is referenced by:  cdlemc3  29650
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616
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